I'm probably late with my advice, but anyway.
A strictly hut-to-hut kike from Interlaken to Zermatt is, no doubt, possible, but would take a lot of time. You have to cross the Bernese range of the Alps, then the Rhone Valley (at the bottom of which there are cities and hotels, but no huts).
So the reasonable thing is to take shortcuts. Use the map link suggested by Arno in post #2 of thread. Go to Mürren by train and start the hike there.
Day 1 Mürren - Griesalp via Sefinenfurgge Pass. Night at Friends of Nature hut Gorneren. There are other possibilities, such as the Rotstockhütte if you want Day 1 to be a short hiking day.
Day 2 Griesalp - Kandersteg. Night at a hotel. From there I suggest you take the train to Hohtann on the Wallis side of the Bernese Alps. Alternatively, you could hike from Kandersteg to the Lötschenpasshütte, but as you can see on the map, the continuation from there to further south is long and in my opinion not so exciting. But possible, then add two days.
Day 3 Hohtann - Ausserberg. I am not aware of any hut or hotel at Hohtann, so I suggest you spend the night at Kandersteg and take a train to Hohtann (one train every hour). Hike the scenic "Südrampe Trail" to Ausserberg, and spend the night there or go on to Randa in the Zermatt Valley by train via Visp. Crossing the Rhone Valley and hiking up the lower part of St-Niklaus Valley is from a hiker point of view not so interesting.
Day 4 Randa - Europahütte. You'll see warning signs on the map, due to landslides some trails are currently closed, but there is access from Randa to Europahütte.
Day 5 Europahütte - Zermatt. Again, the main trail is closed, but there is an alternative trail, it seems.
In Swiss alpine huts, you normally sleep on a mattress and blankets are available. So you need a cloth (silk?) sleeping bag. A thermo sleeping bag and a pad are not necessary. Book in advance. You can normally get dinner, breakfast, and tea for your water bottle, but cannot buy picknick food so have all you need in advance. There are grocery stores in Mürren, Kandersteg, Ausserberg, Randa. In huts you should have cash with you. My advice is don't do this alone, and if the weather is bad you should really follow that advice!
Take the minimum luggage with you and ship the rest from Interlaken to Zermatt, either by post to your hotel in Zermatt, or, rather, from station to station by the railways (you may need to have a ticket, but your railpass should be probably good enough).
Sorry again if I am late with that, but I'm sure you'll have an exciting time anyway!